We consider the problem of evolving nonlinear initial data in the close limit
regime. Metric and curvature perturbations of nonrotating black holes are
equivalent to first perturbative order, but Moncrief waveform in the former
case and Weyl scalar $\psi_4$ in the later differ when nonlinearities are
present. For exact Misner initial data (two equal mass black holes initially at
rest), metric perturbations evolved via the Zerilli equation suffer of a
premature break down (at proper separation of the holes $L/M\approx2.2$) while
the exact Weyl scalar $\psi_4$ evolved via the Teukolsky equation keeps a very
good agreement with full numerical results up to $L/M\approx3.5$. We argue that
this inequivalent behavior holds for a wider class of conformally flat initial
data than those studied here. We then discuss the relevance of these results
for second order perturbative computations and for perturbations to take over
full numerical evolutions of Einstein equations.Comment: 7 pages, 7 figure

Lousto, C. O.

English

arXiv

We consider the problem of evolving nonlinear initial data in the close limit regime. For exact Misner initial data (two equal mass black holes initially at rest), metric perturbations evolved via the Zerilli equation suffer from a premature breakdown (at a proper separation of the holes L/M≈2.2) while we find that the exact Weyl scalar ψ4 evolved via the Teukolsky equation keeps very good agreement with the full numerical results up to L/M≈3.5. Metric and curvature perturbations of nonrotating black holes are equivalent to first perturbative order, but the Moncrief waveform in the former case and the Weyl scalar ψ4 in the latter differ when nonlinearities are present. We argue that this inequivalent behavior holds for a wider class of conformally flat initial data

Lousto, C.

MPG.PuRe

PHYSICAL REVIEW D, VOLUME 63, 047504Perturbative evolution of nonlinear initial data for binary black holes:Zerilli versus Teukolsky equationCarlos O. LoustoAlbert-Einstein-Institut, Max-Planck-Institut fu¨r Gravitationsphysik, Am Mu¨hlenberg 1, D-14476 Golm, Germanyand Instituto de Astronomı´a y Fı´sica del Espacio–CONICET, Buenos Aires, Argentina~Received 17 June 1999; revised manuscript received 7 October 1999; published 30 January 2001!We consider the problem of evolving nonlinear initial data in the close limit regime. For exact Misner initialdata ~two equal mass black holes initially at rest!, metric perturbations evolved via the Zerilli equation sufferfrom a premature breakdown ~at a proper separation of the holes L/M’2.2) while we find that the exact Weylscalar c4 evolved via the Teukolsky equation keeps very good agreement with the full numerical results up toL/M’3.5. Metric and curvature perturbations of nonrotating black holes are equivalent to first perturbativeorder, but the Moncrief waveform in the former case and the Weyl scalar c4 in the latter differ whennonlinearities are present. We argue that this inequivalent behavior holds for a wider class of conformally flatinitial data.DOI: 10.1103/PhysRevD.63.047504 PACS number~s!: 04.30.Db, 04.25.Nx, 04.70.2sThere has been a revival of interest in the perturbationtheory of black holes since the work of Price and Pullin @1#who put forward the close limit approximation. This ap-proach considers the final merger stage of binary black holesas a single, perturbed black hole. They studied the Misnerproblem, two equal mass black holes initially at rest, andcompared their results with full numerical evolution of theEinstein equations for various initial separations. The im-pressive agreement, even for not so small separations, trig-gered several researchers to test these ideas for initial datarepresenting boosted, single spinning plus Brill waves andorbiting black holes @2#.Misner @3# found a solution to the conformally flat, timesymmetric initial value problem representing two black holesat rest separated by a proper distance L parametrized by m0~see below!. For m0,1.8 (L/M,3.3) a common event ho-rizon encompasses the system, and for m0,1.36 (L/M,2.5) a common apparent horizon appears. The Misnerthree-geometry takes a conformal factor F(r ,u)4 whereF5H 11d (nÞ0@~11d2!sinh2 nm01d cos u sinh 2nm01d2#21/2J Y S 11 M2R D , ~1!where d5M /(4RS1), and S15˙ (n51‘ 1/sinh nm0. Here weidentified R, the conformal space radial coordinate, with theSchwarzschild isotropic coordinate, R5˙ 14 (Ar1Ar22M )2.This metric represents an asymptotically flat three-geometry with total Arnowitt-Deser-Misner ~ADM! mass M.The proper distance L between the throats can be written asL5M2S1S 112m0 (n51‘nsinh nm0D . ~2!0556-2821/2001/63~4!/047504~4!/$15.00 63 0475Abrahams and Price @4# found that if one does not linear-ize Misner initial data, but extracts the l multipoles from theexact three-metric, the close limit approximation breaksdown for much smaller separations, in a regime where thelinearized approach precisely agrees with full numerical cal-culations. To linearize the three-metric ~1! one can use theperturbative notion of F421!1. In this case, after expan-sion into Legendre polynomials, one getsF4’118S 11 M2R D(l52,4, . . .k l~m0!~M /R ! l11Pl~cos u!,~3!where k l(m0)5˙ 1/(4S1) l11(n51‘ (coth nm0)l/sinh nm0 . Notethat the expansion parameter, Mcoth(m0)/(4RS1), has to be,1 for the above expansion to be valid. This condition isalways satisfied outside the effective single hole event hori-zon, located at R5M /2, for m0,1.56865. For larger valuesof m0 one has an inverse expansion analogous to that madein Eq. ~2.21! of Ref. @5#.Since to first order we only have quadrupolar (l52) con-tributions, the expansion ~3! has the nice feature of separat-ing, in the common factor k2, all the dependence on theinitial distance between holes. This allows us to compute thewhole family of evolutions with only one integration of theZerilli equation, and then rescale the waveforms by theircorresponding k2(m0). In Regge-Wheeler notation one findsthat the only nonvanishing components of the perturbedthree-metric are H25K . One then builds up the initial valueof the Moncrief waveform @14#, Ql ~with ] tQl50) andevolves to obtain waveforms, spectra, and radiated energies@1#.It was first found by Abrahams and Price @4# that if onedoes not linearize the initial data, but extracts its multipolesfrom the exact metric asH2(l0)5K (l0)5E du sin uF4Y l0 , ~4!©2001 The American Physical Society04-1BRIEF REPORTS PHYSICAL REVIEW D 63 047504and evolves using the Zerilli equation, one obtains for theradiated energy the disturbing results reproduced in Fig. 1.While for small separations L/M,2, corresponding to m0,1.0, the agreement of full numerical computations @6# andlinearized perturbations is good; the energy curve presents alocal maximum at m0’1.25 and then decreases to a localminimum at m0’1.41. At this point the system radiates anorder of magnitude less than the evolution of linearized ini-tial data. For larger values of m0 the radiated energy rapidlyincreases, and soon after m0’1.49, it overestimates the totalenergy radiated by several orders of magnitude. All this hap-pens well before the linearized theory begins to deviate fromthe full numerical results (m0.1.8). To unravel this para-doxical result we will first compute the corresponding radi-ated energies in an alternative formulation of the black holeperturbations.There is an independent formulation of the perturbationproblem derived from the Newman-Penrose formalism @7#that fully exploits the null structure of black holes allowingone to uncouple Einstein equations ~and Bianchi identities!for a single wave equation to describe perturbations aroundKerr black holes. The two approaches are related and equiva-lent when one deals with first order perturbations @8,9#. Herewe want to explore how they behave under nonlinear com-ponents present in the initial data. The outgoing gravitationalradiation is fully described in this gauge ~and tetrad! invari-ant formalism by the Weyl scalar c452Cabgdnam¯ bngm¯ d,where nm and m¯ m ~together with lm and mm) form the tetradthat spans the spacetime.The first step towards building up the initial c4 and ] tc4is to find an instantaneous exact tetrad compatible with thedata ~1!. We have found it by choosing the lm and nm thatgenerate shear free null congruences ~spin coefficients s505l) and fix the form of mm and its complex conjugatem¯ m under transformations of type III ~boosts! in such a wayFIG. 1. The solid curve represents the total radiated energy ~E!computed via the Zerilli equation for two equal mass holes. At thebottom of the dip (L’2.58) the system radiates one order of mag-nitude less energy than that predicted by the evolution of linearizedinitial data ~dotted line!.04750that the spin coefficient e50. Our tetrad is then~ lm!5S 1122M /r ,F22,0,0 D ,~nm!512 1,2~122M /r !F22,0,0,~mm!5F22A2r~0,0,1,i/sinu!. ~5!Using formulas ~3.1! and ~3.2! of Ref. @10#, which givec4 and ] tc4 in terms of the three-geometry and the extrinsiccurvature ~here vanishing!, we obtainc4u t505~122M /r !2r2F6$23~]uF!21F~]u22cot u]u!F%,] tc4u t50522Mr2F2c4Ut50. ~6!The evolution of these initial data via the Teukolsky equa-tion produces the results shown in Fig. 2 for the total energyemitted as gravitational waves. There is no dip at any valueof the separation of the holes. The predicted energy agreeswith the full numerical results for all m0,1.8. For largervalues of the separation ~although we evolved here exactinitial data for two distinct black holes! we overestimate theradiated energy at practically the same rate as the originalresult of Price and Pullin who extrapolated the linearized, l52, initial data to values of (m0.1.58) beyond the radius ofconvergence of their expansion parameter, i.e.,M coth(m0)/(4RS1).1. Correcting for this fact produces thel52 piece of the curve labeled as ‘‘Linear.’’FIG. 2. The solid line gives the total radiated energy ~E! ascomputed using the Newman-Penrose-Teukolsky approach. Thereis complete agreement with full numerical results for L,3.3 andwith the Price-Pullin results ~labeled as PP, l52) all the way up tomuch larger initial separations. Also shown are two other notions ofperturbative initial data in the Newman-Penrose formalism.4-2BRIEF REPORTS PHYSICAL REVIEW D 63 047504We also essayed two possible definitions of the perturba-tive notion in the Newman-Penrose formalism. The first one,which we called perturbative, considers the exact Weyl ten-sor contracted with the background ~Schwarzschild! tetradand a second possibility, which we called linear, considersonly linear terms in the conformal factor F . The resultingenergy is essentially unchanged in the m0,1.8 regime andgrows steeper than the exact initial data for larger initialseparations of the holes. This is because for the initial per-turbative c4’s higher l contributions are less under controlthan for the exact initial c4. This is seen in the waveformsextracted far away from the system. While for the perturba-tive choices of c4 waveforms look fine up to m’1.6 thoseevolved from the exact c4 reach m’2.0 without muchhigher l content. No dip was found at any value of the sepa-ration of the holes and in fact we could not reproduce theresults of Fig. 1 within the Newman-Penrose-Teukolsky for-malism. This gives us a measure of a certain robustness ofthis approach against nonlinearities included in the initialdata.Since both evolution equations, Zerilli’s and Teukolsky’s,are equivalent in the linear perturbations regime, the differ-ent results obtained must be related to how c4 and Q handlethe nonlinearities included in the Misner data. The quantitydirectly related to the radiation ] tQ initially vanishes forMisner data ~time symmetric!; in Fig. 3, we plot ] t2Q at t50. We observe a good superposition for r*.0. There isalso a big depression for linear data with a minimum atr*/2M’23. As the two black holes start from larger initialseparations ~m0 increases!, the amplitude of this depressiondecreases and its location recedes towards more negativer*’s. This continues thus until we reach m0’1.41. Then theamplitude begins to grow while still the location of the de-pression recedes towards negative r*’s. Is this relativelyFIG. 3. The initial second time derivative of the Moncrief wave-form normalized by k2 to compare with the linear initial data.While there is a good superposition of the data for r*.0, the lo-cation of the maximum amplitude monotonically recedes towardsmore negative r* as m0 increases. This provokes a decrease of thefrequency for which destructive interference of the outgoing radia-tion occurs.04750small decrease of the amplitude ~around 20%! responsiblefor one order of magnitude less radiated energy? To answerthe above question one has to take into account the wavenature of the gravitational radiation. We know that a greatdeal of the radiation reaching infinity is generated around themaximum of the Zerilli potential at rmax* /2M’0.95. Therewill also be a piece of the disturbance generated at aroundthe pick of ] t2Q . These two pulses will be out of phase by(2Mv)(Dt/2M ) where Dt is the time the pulse generatedclose to the horizon takes to arrive at r*/2M’0.95. Depend-ing on their relative phases these two pulses can produceconstructive or destructive interference. Assuming Dt;Dr*, where Dr* is the distance between the pick of ] t2Qand the maximum of the Zerilli potential, this analysisshows, that in the linear regime the destructive interferenceappears for frequencies 2Mv.1.2, too high to influence thetotal energy radiated at infinity. As we increase m0, the pickin Fig. 3 moves to the left, thus generating destructive inter-ference at lower frequencies. In Fig. 4 we show that thiseffect is clearly visible in the three spectra for m051.38,1.41, 1.45. There is destructive interference around 2Mv’0.95, 0.83, 0.65, respectively. The strong suppression ofthe radiation at m051.41 is then due to destructive interfer-ence right at the frequency of the maximum of the linearspectrum, at 2Mv;0.8. By the same mechanism we canexplain the sudden increase in the radiated energy for m0.1.4. It is now the effect of destructive interference acting attoo low frequencies and constructive interference at higherones together with a dramatic increase of the amplitude ofthe initial data with respect to the linear regime. Finally form0.1.56865 the effect of the change in the perturbative pa-rameter makes things blow up.There remains the question of why nothing like this hap-pens when we use the Newman-Penrose-Teukolsky approachFIG. 4. The spectrum of the gravitational radiation as evolvedby the Zerilli equation. The solid line labeled with the separationparameter m051 gives essentially the linear regime ~we rescaled itsamplitude by one-fifth!. We see how the suppression effect sets upsharply around m051.41. For m051.38, 1.41, 1.45 destructive in-terference occurs around frequencies 2Mv’0.95, 0.83, 0.65, re-spectively.4-3BRIEF REPORTS PHYSICAL REVIEW D 63 047504to Schwarzschild perturbations. Again, a plot of the initialdata for different separations leads to the answer. We ob-serve that the maximum of the initial data for increasing m0shifts toward increasing r*’s instead of more negative onesas happened for the Zerilli formalism. Thus regardless of asmall decrease in the amplitude the destructive interferenceoccurs at too high frequencies to influence the outgoing ra-diation.Our results are relevant to the idea of evolving fully nu-merically binary black holes starting from large separations~starting from post-Newtonian initial data! until a commonhorizon encompasses the system and then let perturbationtheory to take over @11#. The perturbation taking over the fullnumerical method has the advantage of optimizing super-computer resources, concentrating them in the region wherethe two black holes are completely detached and full nonlin-ear relativistic effects take place. Once a common horizonforms one can assume the close limit approximation to holdand continue the evolution with a single wave equation onthe background of a Kerr black hole. It is very fortunate thatis the Newman-Penrose-Teukolsky approach rather than theRegge-Wheeler-Zerilli one that has this nice behavior in re-sponse to nonlinear Cauchy data since the Teukolsky equa-tion can be generalized to rotating black hole backgroundswhile the metric perturbation approach cannot. The effectdescribed in Fig. 1 inhibits us from using the Zerilli equationto that end. On the other hand, the Teukolsky evolutionseems better suited for this marriage between full numericaland perturbative approaches. We also checked that the samegeneral bad behavior of the Zerilli equation and healthy onefor the Teukolsky equation hold when one considers Brill-04750Lindquist initial data ~other solutions of the conformally flat,time symmetric initial value problem!. This behavior is alsotrue for orbiting black holes if we remain within the Bowen-York family of initial data ~conformally flat and longitudi-nal!. In this case we can make a decomposition of the con-formal factor of the type @12# F5FMisner1Freg , whereFreg is proportional to the square of the momentum of theholes and the square of their distance. This means that atleast for orbiting black holes with small initial momentumthe effects discussed in this paper should qualitatively still bepresent.Another situation where our results should be consideredis when one is interested in studying second order perturba-tions of rotating black holes @13#. The perturbative approachto the Newman-Penrose formalism forms a hierarchy ofequationsTˆc4(N)5S@c (N21),] tc (N21)# , ~7!where T stands for the background wave operator of the Teu-kolsky equation, c4(N) is the waveform of the N perturbativeorder considered, and S is a source term formed by productsof all perturbations of order lower than N. One can solve theabove equations by evolving the initial data order by order,successively reaching the next perturbative stage. An alter-native approach to that is to evolve exact initial data with thefirst order wave equation ~it has a vanishing source term invacuum!, and then evolve second order equations with van-ishing initial data ~expressing the source in terms of the firstorder perturbations!. This is consistent with the desired sec-ond perturbative order.@1# R. Price and J. Pullin, Phys. Rev. Lett. 72, 3297 ~1994!.@2# J. Pullin, in Proceedings of the GR15, edited by N. Dadhichand J. Narlikar ~Inter-University Centre for Astronomy andAstrophysics, Puna, 1998!, p. 87.@3# C. W. Misner, Ann. Phys. ~N.Y.! 24, 102 ~1963!; R. W.Lindquist, J. Math. Phys. 4, 938 ~1963!.@4# A. Abrahams and R. Price, Phys. Rev. D 53, 1963 ~1996!.@5# C. O. Lousto and R. Price, Phys. Rev. D 55, 2124 ~1997!.@6# P. Anninos, D. Hobil, E. Seidel, L. Smarr, and W.-M. Suen,Phys. Rev. D 52, 2044 ~1995!.@7# S. A. Teukolsky, Astrophys. J. 185, 635 ~1973!.@8# M. Campanelli and C. O. Lousto, Phys. Rev. D 58, 024015~1998!.@9# M. Campanelli, W. Krivan, and C. O. Lousto, Phys. Rev. D58, 024016 ~1998!.@10# M. Campanelli, C. O. Lousto, J. Baker, G. Khanna, and J.Pullin, Phys. Rev. D 58, 084019 ~1998!.@11# J. Baker, B. Bru¨gmann, M. Campanelli, and C. Lousto, Class.Quantum Grav. 17, L149 ~2000!.@12# C. O. Nicasio, R. J. Gleiser, R. H. Price, and J. Pullin, Phys.Rev. D 59, 044024 ~1999!.@13# M. Campanelli and C. O. Lousto, Phys. Rev. D 59, 124022~1999!.@14# V. Moncrief, Ann. Phys. ~N.Y.! 88, 323 ~1974!. .4-4

Perturbative evolution of nonlinear initial data for binary black holes: Zerilli vs. Teukolsky

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